Lipsman mapping and dual topology of semidirect products. (English) Zbl 1421.22005

Rahali, Aymen

Bull. Belg. Math. Soc. - Simon Stevin 26, No. 1, 149-160 (2019).

Let \(G\) be a second countable locally compact group. The set of all equivalence classes of irreducible unitary representations \([\pi]\) of \(G\) is denoted by \(\widehat{G}\), the unitary dual of \(G\). It is known that \(\widehat{G}\) is equipped with the Fell topology. The description of the dual topology is a candidate for some aspects of harmonic analysis on \(G\). Generally for an exponential solvable Lie group \(G\), its dual space \(\widehat{G}\) is homeomorphic to the space of coadjoint orbits by the Kirillov mapping. For every admissible linear \(\psi\) of the dual \(\mathfrak g^*\) of \(\mathfrak g = \mathrm{Lie}(G)\), one can construct an irreducible unitary representation \(\pi_\psi\) by holomorphic induction. Let \(\psi \mapsto \pi_\psi\) the map of the set \(\mathfrak g^\ddagger\) of the admissible linear forms onto the dual space \(\widehat{G}\). It was pointed out by Lipsman that the correspondence: \[ \begin{aligned} \Theta: \mathfrak g^\ddagger/G &\longrightarrow \widehat{G}\\ \mathcal O &\longmapsto \pi_{\mathcal O} \end{aligned} \] is bijective.
In this paper, the author studies the case when \(G= K\ltimes V\) is the semidirect product of a connected compact Lie group \(K \) acting by automorphism on a finite dimensional real vector space \((V, \langle, \rangle)\). The coadjoint orbit of \(G\) passing through \((f, l)\in \mathfrak g ^*\) is given by: \[ \mathcal O_{(f, \ell)}= \{ \mathrm{Ad}^*_K (k)f + k.\ell \odot v, k.l): k\in K, v\in V \}, \] where \(\ell\odot v (A) = \ell(A.v), \forall A \in \mathfrak k= \mathrm{Lie}( K) \).
For \(\Omega \) a \(K\)-orbit in \(V^*\) and an arbitrary \(\ell \in \Omega\), he proves: \[ \widehat{G}= \widehat{K}\bigcup (\bigcup_{\Omega \in \Lambda} \widehat{G}(\Omega)), \quad \text{where } \widehat{G}(\Omega)= \{ \mathrm{Ind}^{G}_{K_\ell\ltimes V}(\rho \otimes \chi_\ell): \rho \in \widehat{K_\ell}\}, \] here \(\Lambda\) is the set of the non-trivial orbits in \(V^*/K\).
Let \(\rho_\lambda \) be an irreducible representation of \(K_\ell\) with highest weight \(\lambda \), the representation of \(G\) obtained by holomorphic induction from \((\mu, \ell)\in \mathfrak g^*\) is equivalent to the representation \(\pi_{(\mu, \ell)}= \mathrm{Ind}^G_{K\ell\ltimes V}(\rho_\mu\otimes \chi_\ell) \). When that \(G\) is exponential, the author proves that the Lipsman mapping \(\Theta: \mathfrak g^\ddagger/G\longrightarrow \widehat{G} \) is a homeomorphism.

Reviewer: Mohamed Selmi (Sousse-Riadh)

MSC:

22D10	Unitary representations of locally compact groups
22E27	Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods
20G05	Representation theory for linear algebraic groups

Keywords:

Lie groups; semidirect product; unitary representations; coadjoint orbits; symplectic